The subschemes \(Y_\beta \)  and \(Z_\beta \)  can be more explicitly defined in terms of weights for the action of a maximal torus \(T < G\)  as follows. The linearisation \(\mathcal {L}\)  (or some positive power) determines a \(G\) -equivariant embedding \(Y \hookrightarrow \mathbb {P}^n\)  and we can choose coordinates on \(\mathbb {P}^n\)  such that the action of the fixed maximal torus \(T \subset G\)  is given by \(t \mapsto \operatorname{diag}(\alpha _0(t), \dots , \alpha _n(t))\)  for characters \(\alpha _i : T \rightarrow \mathbb {G}_m\) ; then by [1]}, we have
\(\overline{Y_\beta } & = Y \times _{\mathbb {P}^n} \lbrace  [p_0: \dots :p_n] \in \mathbb {P}^n : p_i = 0 \text{ if } \alpha _i \cdot \beta < ||\beta ||^2 \rbrace  \nonumber \\{Y_\beta } & = \overline{Y_\beta } \times _{\mathbb {P}^n} \lbrace  [p_0: \dots :p_n] \in \mathbb {P}^n : \exists i \text{ such that } p_i \ne 0 \text{ and } \alpha _i \cdot \beta = || \beta ||^2 \rbrace  \\Z_\beta & = Y \times _{\mathbb {P}^n} \lbrace  [p_0: \dots :p_n] \in \mathbb {P}^n : p_i = 0 \text{ if } \alpha _i \cdot \beta \ne ||\beta ||^2 \rbrace  \nonumber .\) 
